The widespread technological introduction of graphene beyond electronics rests on our ability to assemble this two-dimensional building block into three-dimensional structures for practical devices. To achieve this goal we need fabrication approaches that are able to provide an accurate control of chemistry and architecture from nano to macroscopic levels. Here, we describe a versatile technique to build ultralight (density ≥1 mg cm−3) cellular networks based on the use of soft templates and the controlled segregation of chemically modified graphene to liquid interfaces. These novel structures can be tuned for excellent conductivity; versatile mechanical response (elastic-brittle to elastomeric, reversible deformation, high energy absorption) and organic absorption capabilities (above 600 g per gram of material). The approach can be used to uncover the basic principles that will guide the design of practical devices that by combining unique mechanical and functional performance will generate new technological opportunities.